Topics in PDE
We read selected chapters from classical textbooks on PDE such as:
The plan is to prepare the reading material in advance and then discuss the material and exercises during the meetings.
In this Cluster Group, the fundamental principles of Kolmogorov and Fokker-Planck equations are explored, which are indispensable tools for comprehending stochastic processes and the dynamics of continuous systems influenced by randomness. The seminar encompasses their theoretical foundations and applications.
Key topics to be covered include:
Initially, we will delve into the classical theory, bridging the gap between stochastic processes and Partial Differential Equations (PDEs). Subsequently, we aim to explore a neoclassical approach, including the use of the fractional Laplacian as an illustrative example.
Based on Ros-Oton–Serra's The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Silvestre's Regularity of the obstacle problem for a fractional power of the laplace operator and Chen–Song's Estimates on Green function and Poisson kernels for symmetric stable processes.
Reading sources included Evan's Partial Differential Equations, Brezis' Functional Analysis, Sobolev Spaces, and Partial Differential Equations, Stein's Singular Integrals and Differentiability Properties of Functions and Duistermaat and Kolk's Distributions.
Reading Introduction to Riemannian manifolds by John M. Lee, supplemented by Heat kernel and analysis on manifolds by Alexander Grigor'yan.
Reading Spectral graph theory by Fan R. K. Chung
Discuss Lévy Processes and Infinitely Divisible Distributions by Ken-iti Sato
Discuss